A210533 Number of ways to write 2n = x+y (x,y>0) with x-1 and x+1 both prime, and x and x^3+y^3 both practical.

0, 0, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 1, 5, 3, 5, 5, 5, 3, 6, 5, 6, 6, 6, 3, 6, 2, 7, 7, 6, 6, 7, 1, 6, 8, 8, 3, 8, 4, 6, 8, 7, 4, 8, 4, 8, 8, 6, 5, 8, 5, 6, 9, 7, 3, 9, 6, 8, 9, 8, 5, 9, 3, 7, 9, 7, 5, 9, 2, 7, 9, 7, 4, 10, 4, 8, 10, 8, 5, 10, 8, 7, 10, 10, 6, 10, 4, 9, 11, 8, 7, 11, 6, 11, 12, 11

Offset: 1

Author

Zhi-Wei Sun, Jan 28 2013

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>2. Moreover, for each m=2,3,4,... any sufficiently large even integer can be written as x+y (x,y>0) with x-1 and x+1 both prime, and x and x^m+y^m both practical.

Examples

			a(17)=1 since 2*17=12+22 with 11 and 13 both prime, and 12 and 12^3+22^3=12376 both practical.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Crossrefs

Cf. A005153, A210528, A210531, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320.

Programs

Mathematica

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[PrimeQ[2k-1]==True&&PrimeQ[2k+1]==True&&pr[2k]==True&&pr[(2k)^3+(2n-2k)^3]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]